### 2.14 The role of diffeomorphisms

As in other discrete formulations, the spatial diffeomorphism group cannot be realized exactly on the
lattice, and the only obvious symmetries of a cubic lattice are discrete rotations and overall translations.
The commutator computation described above indicates that one may be able recover the diffeomorphism
invariance in a suitable continuum limit. Corichi and Zapata [83] have suggested the presence of a residual
diffeomorphism symmetry in the lattice theory, under which, for example, all non-intersecting Wilson loop
lattice states would be identified.
One can try to interpret the lattice theory as a manifestly diffeomorphism-invariant construction, with
the lattice representing an entire diffeomorphism equivalence class of lattices embedded in the
continuum [146]. In order to make this interpretation consistent, one should modify the functional form of
either the Hamiltonian or the measure, in such a way that the commutator of two lattice Hamiltonians
vanishes, as .